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An article based on two papers that are published this month in the Proceedings: paper 4312 S entitled ‘The effect of a back-e.m.f. nonlinearity on responses of a secondorder position control system’ by F. Fallside andM. R. Patel, and paper 4273 S entitled ‘Application of the second method of Lyapunov to the stability of certain position control systems containing a back-e.m.f. nonlinearity’ by F. Fallside and J. O. C. Ezeilo. The papers will be read before a meeting of the Science and General Division on the 9th December 1963 at Savoy Place. Dr. Fallside and Mr. Patel are in the Department of Engineering, University of Cambridge, and Dr. Ezeilo is in the Department of Mathematics, University of Ibadan.

The axial instability of winding conductors is one of the frequent modes of short-circuit failures in power transformers. This paper throws a new light on this failure mode for helical and barrel coils and takes important steps forward in three ways: (i) recognising a new mode of possible failure. (ii) accounting for the dynamic considerations leading to the formula for calculating the mechanical resonance frequency which one must avoid in the design. (iii) calculating the critical strength of winding conductors which should not be exceeded. The conclusions here will help the design engineer to keep the transformer windings stable under short-circuit forces.

In the linearized analysis of control systems containing a d.c. field-excited motor the armature current is assumed constant, allowing motor torque to be taken proportional to field current. Since, in practice, this introduces the complication of current forcing to overcome variations caused by motor back-e.m.f., the question arises of how much armature current forcing, or constancy, is necessary to give desirable system responses. The variation introduces a two-variable (field current, motor speed) non-linearity into the system equation. The paper investigates the effect of the non-linearity on the step function and ramp and frequency responses of an otherwise ideal output-velocity-stabilized second-order position control system. Increase of the non-linearity is initially similar to increased system damping until a heavily non-linear and undesirable type of response is reached; unstable ramp responses can occur. There is, however, a well defined boundary marking the onset of the latter responses; below it responses are comparable with linear, particularly for critical or heavy damping. The range of responses is investigated, together with armature-current behaviour during them. This, together with the boundaries, gives a specification of armature supply requirements for a desirable response. The effects of load viscous friction and field-current saturation are also investigated.

A Lyapunov function, for which both V and V are sign-definite in only certain regions of the phase plane, is used to estimate the region of asymptotic stability of the equilibrium point of a nonlinear control system. An analytical result is obtained by restricting attention to a subregion where V and V are signdefinite.

It is shown that, under certain conditions, a speed-control system with a back-e.m.f. nonlinearity fails to reach the demanded speed in response to a step-function input. This is equivalent to a restriction on the region of asymptotic stability of the steady state of the system. The restriction is investigated by means of the Lyapunov method, and an analytical expression is obtained for it. The nonlinearity is a function of two variables, and the functions V and V used are not sign-definite in the entire phase plane. A technique is used for deriving analytical results in this situation, and the resulting sufficient conditions are compared with exact results.